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332 ALGEBRA. Now the event is certain to happen or to fail; therefore the sum of the chances of happening and failing must represent certainty. If therefore we agree to take certainty as our unit, we have

$$1 = k(a + b)$$, or $$k = {1 \over a+b}$$, ∴ the chance that the event will happen is $a⁄a+b$, and the chance that the event will not happen is $b⁄a+b$.

Cor. If $$p$$ is the probability of the happening of an event, the probability of its not happening is $$1-p$$.

404. The definition of probability in Art. 402 may be given in a slightly different form which is sometimes useful. If c is the total number of cases, each being equally likely to occur, and of these a are favorable to the event, then the probability that the event will happen is a c, and the probability that it will not happen is 1-a c.

Ex. 1. (a) From a bag containing 4 white and 5 black balls a man draws a single ball at random. What is the chance that it is black ?

A black ball can be drawn in 5 ways, since any one of the 5 black balls may be drawn. In the same way any one of the 4 white balls may be drawn.

Hence the chance of drawing a black ball is $5⁄4+5$, or $5⁄9$.

(b) Suppose the man draws 3 balls at random. What are the odds against these being all black ?

The total number of ways in which 3 balls can be drawn is 9C3, and the total number of ways of drawing 3 black balls is 5C3; therefore the chance of drawing 3 black balls

$$={5C3 \over 9C3} = {5.4.3 \over 9.8.7} = {5 \over42}.$$

Thus the odds against the event are 37 to 5.